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### Why There are 12 Tones in a Scale

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Date: 12/28/2000 at 18:52:43
From: Ashley
Subject: Why 12 tones in a scale

I've written to you before, but I need help again. Your last answer
was very helpful and I'm grateful for it!

I have a question about some information I got from another source as
to why there are 12 tones in a scale. We choose an r, which is some
number that is close to 1.5. To get the 12 notes of the scale,
multiply one frequency by r and keep multiplying what you get by r.
Then, bring the frequencies down into the appropriate octave by
dividing by 2 as many times as needed. This r should be close to 3/2
(1.5) since this is a perfect fifth. However, a perfect fifth doesn't
work because you never get an octave that gives a 2:1 ratio with the
note that you started with after you put your ending note down into
the appropriate octave.

I got the equation r^n = 2^m, where n = the number of fifths needed to
complete the progression (which is also the number of notes in the
scale), and m = the number of octaves needed to get to the end of the
progression.

David Rusin's Web page says that the trick is to get an r that is
close to 1.5.

Mathematics and Music - David Rusin
http://www.math.niu.edu/~rusin/uses-math/music/

He says that there is no better value for n than 12 until you get to
29. To fill in the equation and get the value of r, you need to have
the value for m, the number of octaves used.

Is there an easier way to get the number of octaves? One that can be
done without multiplying a frequency by 1.5 many times and then
dividing repeatedly by 2? Also, can you explain why r^n = 2^m? Why
should this be? What's the reasoning behind it? Also, what's so
special about a fifth? Why should the scale be based on the fraction
3/2? I'm doing a project for school that's due when I come back from
Christmas break and I have to teach it to the class, so I'm trying to
questions that you can and it will be greatly appreciated - you're a
great help!
```

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Date: 12/29/2000 at 16:51:37
From: Doctor Rick
Subject: Re: Why 12 tones in a scale

One question you asked is why the ratio 1.5 is important. It was
Pythagoras, I believe, who noticed that two vibrating strings make a
harmonious sound together if the ratio of their lengths is a ratio of
small whole numbers. The frequencies of the resulting sounds are
inversely proportional to the lengths of the strings, so these ratios
will again be ratios of small whole numbers.

The simplest such ratio between different tones is 2:1, which is the
ratio of tones an octave apart. The next simplest is 3:1. Here the
second tone is more than an octave above the first; if we divide it by
2 to bring it down an octave, we get the ratio 3:2, or 1.5:1.
Recognize the number? This second-simplest ratio is what we call a
fifth.

You ask where the equation comes from. Because it is ratios that
matter in musical pitches, we'd like to build a scale as a geometric
sequence of frequencies, in which the ratio between successive
elements of the sequence is a constant. The octave of every tone (the
tone with twice the frequency) must be in the scale, because that's
the simplest, best-sounding ratio. Thus we'll have n tones in an
octave, and the tones in any other octave will be just these n tones
multiplied by a power of 2.

We'd like to be able to build the scale in such a way that, for any
tone in the scale, its fifth (that is, 1.5 times its frequency) is
also in the sequence. This allows you to make pleasant-sounding chords
in any key - that is, you can build a chord starting with any tone in
the scale.

We can build this scale by starting with the tonic (ratio 1) and
finding its fifth (1.5 * 1 = 1.5), then the fifth of this tone (1.5 *
1.5 = 2.25), then the fifth of this tone (2.25 * 1.5 = 3.375), etc.
For each of these tones, its octaves will also be in the scale: you
can multiply or divide the ratio by 2 to get other tones in the scale,
and particularly one that is in the range 1 to 2. The fifth of the
fifth is thus one octave above 2.25/2 = 1.125, and the fifth of this
is an octave above 3.375/2 = 1.6875.

We want this process to terminate sooner or later, otherwise we'll
have an infinite number of tones in our scale. How can it terminate?
When we get to a tone that we have already found. This will happen
when we get a tone that is a power of two: this tone is the same as
the starting tone, some number m of octaves up. If it's the nth tone
we've found, then we have:

1.5^n = 2^m

Unfortunately, there is no solution to this equation with whole
numbers for n and m. Instead, we look for a ratio r that is close to a
fifth (1.5) but is compatible with having the octaves in the scale.
This is where we get the equation:

r^n = 2^m

The other question you're asking is about the math: How can you find
an r that makes this work, with relatively small numbers n and m? You
don't have to use trial and error, at least not so much of it. You can
solve the equation by taking the log of both sides:

log(r^n) = log(2^m)

n*log(r) = m*log(2)

m/n = log(r)/log(2)

Put r = 1.5, and the right side evaluates to 0.584962500... What we
need to do is to find a number m/n that is close to this number, where
m and n are the smallest possible whole numbers. You can try different
values of n (the number of tones in the scale), making a table of:

m = n*log(1.5)/log(2)

= r*0.584962500

The values of m will not be whole numbers. Round each m value to the
nearest whole number, then use this rounded value to compute:

r = 2^(m/n)

You will find that r is close to 1.5 when n = 12, and again when n =
24. A 24-tone scale is just the 12-tone scale with 12 more
quarter-tones in between. The next value for n that gives a better
approximation to the fifth is 29 tones.

I hope this information helps you.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 03/13/2006 at 22:08:34
From: Gin
Subject: why 12 tones in a scale

Doctor Rick said that "the next value for n that gives a better
approximation to the fifth is 29 tones."  Please explain.  I thought
I understood why 12 and 24 tones gives an approximation to the fifth,
but 29 has me wondering as I assumed the next number would be 48.
```

```
Date: 03/14/2006 at 18:07:11
From: Doctor Rick
Subject: Re: why 12 tones in a scale

Hi, Gin.

on this topic than you probably want to know! A brief summary is
found here:

Let's do some of what David Rusin said to do. I just made a
spreadsheet to look at the ratios for 5 through 42 tones per octave.
Here is how it starts:

5 1 1.1486 1.3195 1.5157 1.7411 2

6 1 1.1224 1.2599 1.4142 1.5874 1.7817 2

7 1 1.1040 1.2190 1.3459 1.4859 1.6406 1.8114 2

8 1 1.0905 1.1892 1.2968 1.4142 1.5422 1.6817 1.8340 2

9 1 1.0800 1.1665 1.2599 1.3607 1.4697 1.5874 1.7144 1.8517 2

10 1 1.0717 1.1486 1.2311 1.3195 1.4142 1.5157 1.6245 1.7411 1.8660 2

11 1 1.0650 1.1343 1.2080 1.2866 1.3703 1.4594 1.5544 1.6555 1.7631
1.8778 2

12 1 1.0594 1.1224 1.1892 1.2599 1.3348 1.4142 1.4983 1.5874 1.6817
1.7817 1.8877 2

Then I looked for the tone in each scale that comes closest to 1.5,
which is in the second column below:

5   1.515716567   0.015716567
6   1.587401052   0.087401052
7   1.485994289   0.014005711
8   1.542210825   0.042210825
9   1.469734492   0.030265508
10   1.515716567   0.015716567
11   1.459480106   0.040519894
12   1.498307077   0.001692923
13   1.531966357   0.031966357
14   1.485994289   0.014005711
15   1.515716567   0.015716567
16   1.476826146   0.023173854
17   1.503406654   0.003406654
18   1.527435131   0.027435131
19   1.493758962   0.006241038
20   1.515716567   0.015716567
21   1.485994289   0.014005711
22   1.506195553   0.006195553
23   1.479610446   0.020389554
24   1.498307077   0.001692923
25   1.515716567   0.015716567
26   1.49166449    0.00833551
27   1.50795418    0.00795418
28   1.523239565   0.023239565
29   1.501294382   0.001294382
30   1.515716567   0.015716567
31   1.495517882   0.004482118
32   1.509164428   0.009164428
33   1.490459915   0.009540085
34   1.503406654   0.003406654
35   1.485994289   0.014005711
36   1.498307077   0.001692923
37   1.510048194   0.010048194
38   1.493758962   0.006241038
39   1.504979244   0.004979244
40   1.489677463   0.010322537
41   1.500419433   0.000419433
42   1.510721887   0.010721887

The third column shows the difference between this ratio and 1.5. If
you look down the list, you'll see that multiples often have the same
closest ratio to 1.5: 5, 10, 15, 20, 25, and 30 all have 1.515716567.
In a 35-tone scale, the same value is there, but another becomes
closer to 1.5 than this one.

In the same way, 12, 24, and 36 all have the same closest ratio to
1.5, namely 1.498307077. This is closer to 1.5 than any value before
12, *or* any value after 12 until you get to 29. As I said, it is
the same as the ratio for a 24-tone scale, but the 29-tone scale is
the first that has a ratio *closer* to 1.5 than this (namely,
1.501294382). That's what I meant.

You'll see also that the 29-tone scale is only a little better than
the 12-tone (or 24-tone) scale. However, when you get to a 41-tone
scale, you get a much closer ratio to 1.5, namely 1.500419433.

Mr. Rusin goes into detail elsewhere about how this result can be
found by use of continued fractions, so that it is not necessary to
generate this entire chart. I think that is beyond your need,
however.

I must tell you that I am not far from being out of my depth here; I
am mostly reporting what I read. I'm standing on tiptoe to keep my
nose above water, so don't hold on to me too tightly or I'll go
under. But we can have some fun swimming together if you have more
questions, and there is probably a Math Doctor with more knowledge
of music, continued fractions or whatever, who can assist us if
necessary.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
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